ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS

ZEMER KOSLOFF

Dedicated to the memory of Roy Adler whose work has been and still is a great inspiration for me.

d Abstract. We prove that for every d 6= 3, there is an Anosov diffeomorphism of T which is of stable Krieger type III1 (its Maharam extension is weakly mixing). This is done by a construction of stable 1 type III1 Markov measures on the golden mean shift which can be smoothly realized as a C Anosov 2 diffeomorphism of T via the construction in our earlier paper.

1. Introduction

A topological Markov shift (TMS) on S is the shift on a shift invariant subset Σ ⊂ SZ of the form

n o Z ΣA := x ∈ S : Axi,xi+1 = 1 , where A = {As,t}s,t∈S is a {0, 1} valued matrix on S. A TMS is (topologically) mixing if there exists n n ∈ N such that As,t > 0 for every s, t ∈ S. TMS appear in ergodic theory as a symbolic model for C1+α Anosov and Axiom A diffeomorphisms via the construction of Markov partition of the manifold M [AW, Sin, Bow, Adl]. In this work we will restrict our attention to measures (fully) supported on the Golden Mean Σ ⊂ {1, 2, 3}Z, which is defined by its adjacency matrix   1 0 1   A =  1 0 1  . 0 1 0

2 The TMS Σ is the symbolic space arising from a Markov partition of the automorphism of T given by f(x, y) = (x + y, x) mod 1, See Section 4. The connection between Σ and f was used in [Kos1] for the 1 2 construction of conservative, ergodic C Anosov diffeomorphisms of T without a Lebesgue absolutely continuous invariant measure (a.c.i.m.). Krieger in [Kri] has classified the nonsingular transformations up to orbit equivalence according to the associated flow on the ergodic decomposition of the Maharam extension. The Anosov diffeomorphisms in [Kos1] are of type III1 which means that their Maharam extension is ergodic (the associated flow is a trivial flow). The first step of the construction was to build inhomogeneous Markov measures on Σ such that the shift with respect to (Σ, µ, T ) defined by

(T w)n = wn+1 is a type III1 transformation. After this step, the next step is to utilize the special 1 2 structure of the µ constructed and realize an ergodic C Anosov diffeomorphism g of T which is orbit equivalent to (Σ, µ, T ) hence of type III1. An ergodic nonsingular transformation T is stable type III1 if for any ergodic probability preserving transformation S, T × S is a type III1 transformation. In this paper we construct inhomogeneous Markov measures which are stable type III1. In addition the construction is compatible with the second step (smooth realization) of [Kos1]. As a consequence we get the following result. 1 ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 2

1 d
Theorem. For every d ≥ 4 or d = 2, there exists a C -Anosov diffeomorphism g of T , LebesgueTd which is stable type III1. In particular g is conservative, ergodic with respect to the Lebesgue measure but it has no Lebesgue a.c.i.m.

The reason d 6= 3 is that these diffeomorphisms are Cartesian products of a stable type III1 Anosov 2 d−2 diffeomorphism of T and an hyperbolic linear toral automorphism of T (hence we need d − 2 ≥ 2). The construction of the aforementioned Markov measures uses a reformulation of the stable type

III1 property in terms of measurable equivalence relations and then identifying a special class of holonomies (groupoid elements) of the tail relation which change the tail Radon-Nikodym cocycle in a controlled way. Using these holonomies and elementary Markov Chain arguments we provide an inductive construction of a measure µ such that its Maharam extension is an infinite measure preserving

K automorphism, hence stable type III1, see Corollary 2. The structure of the paper is as follows. Section 2 is an introduction to the relevant material from non-singular ergodic theory, Markov Chains and countable equivalence relations. It provides the definitions and some simple consequences of them. In Section 3 we present the inductive constructions of the aforementioned Markov measures on Σ. Finally in Section 4, we explain how to use the new

Markov measures construction and the smooth realization of [Kos1] to obtain a stable type III1 Anosov 2 diffeomorphism of T .

2. Preliminaries on non-singular ergodic theory and Inhomogeneous Markov Shifts A measurable transformation T of a standard (Polish) measure space (X, B, m) is non-singular if −1 T∗m = m ◦ T and m are equivalent measures. In the case where T is invertible, we denote by n 0 dm◦T n (T ) = dm the sequence of Radon-Nikodym derivatives of T . X = (X, B, m, T ) is ergodic if m A4T −1A
= 0 implies m(A) = 0 or m(X\A) = 0. If T is invertible and the measure m is non atomic then a necessary condition for ergodicity is that the system X is conservative, meaning that there exists no set W ∈ B of positive m measure such that {T −nW } are pairwise disjoint. Sets n∈N with the latter property are called wandering sets. By Halmos recurrence theorem, the system satisfies Poincare recurrence if and only if it is conservative and by Hopf’s criteria X is conservative if and only if for m-almost every x ∈ X ∞ X  0 T k (x) = ∞. k=0 The system K is a K-system if there exists F ⊂ B such that • T −1F ⊂ F . In words F is a factor of X . T∞ −n • F is exact, n=0 T F = {∅,X} modulo measure zero sets. W∞ n • Minimality condition: n=−∞ T F = B. • T 0 is F measurable. The first three conditions are as in the classical setting of ergodic theory (probability preserving maps) whereas the fourth condition comes in order to ensure that X is the unique natural extension of (X, F , m, T ) up to measure theoretic isomorphism [SiT]. A transformation X is weakly mixing if for every ergodic probability preserving transformation (Ω, BΩ, P,S), T × S is an ergodic transformation of (X × Ω, BX ⊗ BΩ, m × P). If T × S is ergodic for all ergodic, invertible ,nonsingular transformation, then X is mildly mixing. If an invertible transformation is mildly mixing then m is equivalent to a T ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 3 invariant probability [FW] while exact (hence non invertible) non singular transformations are mildly mixing [ALW]. ˜ The Maharam extension of an invertible transformation X is a transformation T of (X × R, BX ⊗ BR) 0 defined by T˜(x, y) = (T x, y − log T (x)). It preserves the σ-finite measure defined by for all A ∈ BX , R t ˜ B ∈ BR, µ(A × B) = m(A) B e dt. T is conservative (w.r.t. µ) if and only if T is conservative (w.r.t. m) and (the associated flow on) its ergodic decomposition determines the orbit equivalence class of T

[Kri]. In particular for an ergodic T , T˜ is ergodic if and only if T is of type III1, meaning that its Krieger ratio set is R. See subsection 2.2 for the definition of the ratio set for a cocycle, the Krieger 0 ratio set is the ratio set for ϕ = log T . A transformation X is stable type III1 if for every ergodic probability preserving transformation (Ω, BΩ, P,S), T ×S is type III1. For every probability preserving map S, d(m×P)◦(T ×S) = T 0(x), the Maharam extension of T × S is isomorphic to T˜ × S via the map d(m×P) π(x, w, y) = (x, y, w):(X × Ω) × R → (X × R) × Ω. This observation leads to the following.

Proposition 1. Let X = (X, B, m, T ) be an ergodic non-singular transformation. Then the following are equivalent: (i) T˜ is weak mixing.

(ii) T is stable type III1.

Corollary 2. If T˜ is a conservative K-authomorphism then T is stable type III1 and for every weakly mixing probability preserving transformation (Ω, BΩ, P,S), T × S is stable type III1.

Proof. The first part follows from [ALW, Par], see also [Aar1, Cor 3.1.8.], since a cartesian product of a conservative transformation with a probability preserving transformation is always conservative and a K-automorphism is the natural extension of an exact (hence mildly mixing) factor. For the second part, let R be an invertible, ergodic probability preserving transformation of (Y, C, ν). Since S is weak mixing, then S × R is an ergodic probability preserving transformation. By the stable type III1 property of T , T × (S × R) = (T × S) × R is type III1, showing that T × S is stable type III1. 

The group of automorphisms of a probability space (X, B, m) is the set of all invertible (modm) nonsingular transformations. An action of a countable group Γ on the space (X, B, m) is a map φ :Γ → Aut(X, B, m) which respects the group structure. The notation Γ y (X, B, m) will denote the group action of Γ on X and for γ ∈ Γ, x ∈ X, γx = φγ(x).

2.1. Markov chains.

2.1.1. Basics of Stationary (homogenous) Chains. Let S be a finite set which we regard as the state space of the chain, π = {π(s)}s∈S a probability vector on S and P = (P (s, t))s,t∈S a stochastic matrix. The vector π and P define a Markov chain {Xn} on S by

Pπ,P (X0 = t) = π(t) and Pπ,P (Xn = s |X1, .., Xn−1 ) := P (Xn−1, s) . n P is irreducible if for every s, t ∈ S, there exists n ∈ N such that P (s, t) > 0 and P is aperiodic if for every s ∈ S, gcd {n : P n(s, s) > 0} = 1. Given an irreducible and aperiodic P, there exists a unique stationary (πPP = πP) probability vector πP. ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 4

2.1.2. Inhomogeneous Markov shifts. An inhomogeneous Markov Chain is a stochastic process {X } n n∈Z such that for each times t1, .., tl ∈ Z, and s1, . . . , sl ∈ S, l−1 Y
P (Xt1 = s1,Xt2 = s2,...,Xtl = sl) = P (Xt1 = s1) P Xtk+1 = sk+1 Xtk = sk . k=1
Note that unlike in the classical setting of Markov Chains P Xtk+1 = sk+1 Xtk = sk can depend ∞ on tk. The ergodic theoretical formulation is as follows. Let {Pn}n=−∞ ⊂ MS×S be a sequence of ∞ stochastic matrices on S. In addition let {πn}n=−∞ be a sequence of probability distributions on S so that for every s ∈ S and n ∈ Z, X (2.1) πn−1(t) · Pn (t, s) = πn(s). t∈S Then one can define a measure on the collection of cylinder sets, n o l Z [b]k := x ∈ S : xj = bj ∀j ∈ [k, l] ∩ Z

(here b ∈ SZ) by l−1  l  Y µ [b]k := πk (bk) Pj (bj, bj+1) . j=k Since the equation (2.1) is satisfied, µ satisfies the consistency condition. Therefore by Kolmogorov’s extension theorem µ defines a measure on SZ. In this case we say that µ is the Markov measure generated by {π ,P } and denote µ = M {π ,P : n ∈ }. By M {π, P } we mean the measure n n n∈Z n n Z generated by Pn ≡ P and πn ≡ π. We say that µ is non singular for the shift T on SZ if T∗µ ∼ µ. In order to check if a measure is shift non singular we apply the following reasoning of [Shi], see also [LM].

∞ Definition 3. A filtration {Fn}n=1 of a measure space (X.B) is an increasing sequence of sub σ- ∞ algebras such that the minimal σ-algebra which contains {Fn}n=1 is B. Given a filtration {Fn}, we loc say that ν  µ (ν is locally absolutely continuous with respect to µ) with respect to Fn if for every n ∈ N νn  µn where νn = ν|Fn . The question is when ν loc µ implies ν  µ. Suppose that ν loc µ w.r.t {F }, set z := dνn and n n dµn α (x) := z (x) · z⊕ (x) where z⊕ = 1 · 1 . The sequence {z } is a {F } martingale and n n n−1 n−1 zn−1 [zn−16=0] n n 1 thus ν  µ if and only if {zn} converges in L (satisfies the uniform integrability condition). Theorem 4. [Shi, Thm. 4, p. 528]. If ν loc µ then ν  µ if and only if ∞ X √ [1 − Eµ ( αn| Fn−1)] < ∞ ν a.s. k=1 dν If ν  µ then dµ = limn→∞ zn.

Given a a Markov measure µ = M {πn,Pn : n ∈ Z} on SZ, we want to know when µ ∼ µ ◦ T . The  n Z∩[−n,n] natural filtration on the product space is the sequence of algebras Fn := σ [b]−n; b ∈ S . The measures which we will construct are fully supported on a TMS ΣA meaning that for every n ∈ N, suppPn := {(s, t) ∈ S × S : Pn(s, t) > 0} = suppA. ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 5

loc This implies that µ ◦ T  µ. In addition the measure M {Pn, πn} will be half stationary in the sense that there exists an irreducible and aperiodic stochastic matrix Q such that for every j ≤ 0, Pj := Q and πj = πQ is the unique stationary distribution for Q. This implies for all n ∈ N and Pn−1(xn−1,xn) x ∈ Σ, αn(x) = . Pn(xn−1,xn) By Theorem 4, in this setting, µ is non-singular if and only if ∞ ∞ " # X √ X X p [1 − Eν ( αn| Fn−1)(x)] = 1 − Pn−1 (xn−1, s) Pn (xn−1, s) < ∞ n=−∞ n=0 s∈S for 1 µ ◦ T a.e. x. The following corollary concludes our discussion.

Corollary 5. Let ν = M {πn,Pn : n ∈ Z}, where {Pn} are fully supported on a TMS ΣA and there exists an aperiodic and irreducible P ∈ MS×S such that for all n ≤ 0, Pn ≡ P • ν ◦ T ∼ ν if and only if

∞ X X p p 2 (2.2) Pn (xn, s) − Pn−1 (xn, s) < ∞, ν ◦ T a.s. x. n=0 s∈S • If ν ◦ T ∼ ν then for all n ∈ N, ∞ Y Pk−n (xk, xk+1) (2.3) (T n)0 (x) = . Pk (xk, xk+1) k=0

∞ 2.1.3. A condition for exactness of the one sided shift. Let S be a countable set and {(πn,Pn)}n=1 ⊂ P(S) × MS×S. Denote the one sided shift on SN by σ and by F the Borel σ−algebra of SN. The following is a sufficient condition for exactness (trivial tail σ-field) of the one sided shift which is well known in the theory of non homogenous Markov chains. We include a simple proof.

Proposition 6. Let S be a finite set and µ be a Markov measure on SN which is defined by {πk,Pk : k ∈ N ∪ {0}}. If there exists C > 0 and N0 ∈ N so that for every s, t ∈ S, and k ∈ N,

(2.4) (PkPk+1 ··· Pk+N0−1)(s, t) ≥ C then the one sided shift SN∪{0}, F, µ, σ
is exact.

Remark. In the setting of Markov maps, exactness was proved under various distortion properties [see [Tha] and [Aar1, Chapter 4]]. Their conditions guarantees the existence of an absolutely continuous σ-finite invariant measure. For stationary Markov chains, triviality of the tail algebra (exactness of the shift) was proven by Blackwell and Freedman.

−n Proof. The measure µ ◦ T is the Markov measure generated by Qk := Pk+n and π˜k := πk+n. Let αn n ∗ be the collection of n cylinders of the form [d]1 and α = ∪nαn.

1Here T denotes the two sided (invertible) shift ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 6

n n(B) ∗ For every D = [a]0 ∈ αn and B = [b]0 ∈ α , n(B)−1   Y µ D ∩ T −(n+N0)B = µ(D)(P P ··· P ) P (b , b ) n n+1 n+N0−1 an,b0 N0+n+j j j+1 j=0 n(B)−1 Y −(n+N0) ≥ Cµ(D)πn+N0 (b0) PN0+n+j (bj, bj+1) = C · µ(D)µ ◦ T (B). j=0

Consequently for all B ∈ F and D ∈ αn,   µ D ∩ T −(n+N0)B ≥ C · µ(D)µ ◦ T −(n+N0)(B).

∞ −n −n Let B ∈ ∩n=1σ F and D ∈ αn. Writing Bn ∈ F for a set such that B = T Bn,   −(n+N0) µ (D ∩ B) = µ D ∩ T Bn+N0

−(n+N0) ≥ C · µ(D)µ ◦ T (Bn+N0 ) = Cµ(D)µ(B)

∗ ∗ and thus for every n ∈ N, µ (B |αn ) ≥ Cµ(B). Since αn ↑ α and α generates F,

µ (B |αn )(x) −−−→ 1B(x) µ − a.s. n→∞ by the Martingale convergence theorem. It follows that if µ(B) > 0 then

1B(x) ≥ Cµ(B) > 0 µ − a.s.

∞ −n This shows that for every B ∈ ∩n=1σ F, µ(B) ∈ {0, 1} (the shift is exact). 

2.2. Cocycles and Measurable Equivalence relations . Let (X, B, m) be a standard measure space. A measurable countable equivalence relation on X is a measurable set R ⊂ X × X such that the relation x ∼ y if and only if (x, y) ∈ R is an equivalence relation and for all x ∈ X,

Rx = {y ∈ X :(x, y) ∈ R} is countable. The saturation of a set A ∈ B by the equivalence relation S is the set R(A) := x∈A Rx. We say that R is m−ergodic if for each A ∈ BX , m(R(A)) = 0 or m (X\R(A)) = 0 and R is non-singular if for all A ∈ B, m(A) = 0 if and only if m(R(A)) = 0. An equivalence relation is finite (respectively countable) if for all x ∈ X, Rx is a finite (countable) set. The full group of an equivalence relation R, denoted by [R], is a subgroup of all V ∈ Aut(X, B) such that for m-almost all x ∈ X, (x, V x) ∈ R. A partial transformation of R is a one to one transformation V from B ∈ B to V (B) ∈ B. Denote by Dom(V ) = B the domain of V and Ran(V ) = V (B) the range of V . The groupoid of R, denoted by [[R]], is then defined as the set of all partial transformations of R. Both the full group and the groupoid of R appear in the study of the ergodic decomposition of cocyclic extensions of equivalence relations (and countable group actions). Given an action of a countable group Γ y (X, B, m) one can define the orbit equivalence relation RΓ = {(x, γx): γ ∈ Γ}. By the Feldman-Moore Theorem [Fe-Mo], for every countable equivalence relation R, there exists a nonsingular action on (X, B, m) of countable group Γ for which R = RΓ . Notice that this action is not uniquely defined (there are many such countable group actions). A ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 7 function f :Γ × X → R is an R-valued cocycle of the Γ-action if for all γ1, γ2 ∈ Γ,

f (γ1γ2, x) = f (γ1, γ2x) + f (γ2, x) .

A R-valued orbital cocycle is a function ϕ : R → R such that for all (x, y), (y, z) ∈ R, ϕ(x, z) = ϕ(x, y) + ϕ(y, z).

Two R cocycles ϕ1, ϕ2 are (measurably) cohomologous if there exists and a measurable function b : 0 0 0 0 X → R and a measurable set X ⊂ X with µ (X \ X ) = 0 such that for all (x, y) ∈ (X × X ) ∩ R,

ϕ1(x, y) = ϕ2(x, y) + b(x) − b(y).

An equivalence relation R with a R valued cocycle ϕ define a ϕ-extension of R, which is an equivalence ϕ ϕ relation R ⊂ (X × R) × (X × R), defined by ((x1, y1) , (x2, y2)) ∈ R if and only if (x1, x2) ∈ R and y1 + ϕ (x1, x2) = y2. If Γ y (X, B, m) is such that R = RΓ then every orbital cocycle ϕ : R → R corresponds to a Γ y (X, B, m) cocycle defined (up to set of m measure zero) by f(γ, x) = ϕ (γx, x) [Sch, Prop. 2.3]. The Radon-Nikodym cocycle of a countable equivalence relation R is thus defined dm◦γ by ψRN (x, γx) = log dm (x) where Γ y (X, B, m) is any group acting on X such that R = RΓ. This cocycle defines an orbital cocycle which does not depend (up to a set of measure zero) on our choice of action Γ y (X, B, m) for which RΓ = R. A number r ∈ R is an essential value for an orbital cocycle ϕ : R → R if for all A ∈ B with m(A) > 0 and  > 0, there exists V ∈ [R] such that

m A ∩ V −1A ∩ {x ∈ X : |ϕ(x, V x) − r| < }
> 0.

Write e(R, ϕ) for the set of all essential values of (R, ϕ) . The Krieger ratio set of a nonsingular ϕ transformation X is R(T ) = e (RT , ψRN ). The set e(R, ϕ) is a closed subgroup of R and R is ergodic if and only if R is ergodic and e (R, ϕ) = R. Since in the definition of an essential value one is only concerned with points in A, one can change/relax the definition of the essential values to requiring that V ∈ [[R]] with Dom (V ) ⊂ A. This relaxation, although merely formal, gives the following useful criteria for r to be an essential value for (R, ϕ).

Lemma 7. [CHP, Lemma 2.1.]2 Let R be an ergodic, non-singular, countable equivalence relation of a measure space (X, B, m), ϕ an R cocycle and C ⊂ B a (countable) algebra of sets which generates B. If there exists δ > 0 such that for all  > 0 and C ∈ C , there exists B ⊂ C and V ∈ [[R]] with: (a) Dom(V ) = B and V (B) ⊂ C , (b) m(B) > δm(C) (c) for all x ∈ B, |ϕ(x, V x) − r| <  then r ∈ e(R, ϕ).

Remark 8. The tail relation of non-singular noninvertible transformation (X, BX , ν, S) is defined by

n n TS = {(y1, y2): ∃n ∈ N,S y1 = S y2} .

2See also the formulation in [DaL, Lemma 1.1.] which is similar to the formulation here. Notice that in [CHP], δ = 0.9 yet a similar proof works for a general δ > 0. ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 8

If S is a countable-to-one transformation, then TS is a hyperfinite equivalence relation. In this case −n the trasnformation S is exact if ∩nS B = {∅,X} mod m or equivalently TS is ergodic. For a proof of this statement and the definition of hyperfinite equivalence relations see [Haw].

To every function ϕ : X → R corresponds an orbital cocycle ϕˆ on TS defined by for (y1, y2) ∈ T (S), ∞ X n n (2.5) ϕˆ (y1, y2) := {ϕ (S y1) − ϕ (S y2)} = ϕ (N, y1) − ϕ (N, y2) , n=0 N N where N ∈ N is any number so that S y1 = S y2.

Proposition 9. Given a countable to one non-singular noninvertible transformation (X, BX , ν, S) and ϕˆ a function ϕ : X → R, T (S) = T (Sϕ) where Sϕ is the skew product transformation on X × R defined by Sϕ (x, y) = (Sx, y + ϕ(x)).

Proof. Follows from the definitions. 

If ϕ1 is cohomologous to ϕ2 then e (S, ϕ1) = e (S, ϕ2) [Sch]. If (X, BX , µ, S) is a nonsingular system and ν is a µ equivalent measure then for all γ ∈ [RS], for almost all x ∈ X dν ◦ γ ψ (x, γx) := log (x) RN,ν dν dµ ◦ γ dν dν = log (x) + log (γx) − log (x) dµ dµ dµ

= ψRN,µ (x, γx) + b (γx) − b(x)

dν with b(x) = log dµ (x). This shows that ψRN,ν and ψRN,µ are cohomologous cocycles and implies the following fact.

Fact 10. If (X, BX , µ, T ) is stable type III1 and ν ∼ µ then (X, BX , ν, T ) is stable type III1.

3. Stable type III1 Markov measures For λ > 1 let  λG 1  1+λG 0 1+λG Q =  1 0 1  λ  G G2  0 1 0 √ 1+ 5 where G = 2 is the golden mean. The following relation is the main observation for the inductive construction. Q (3, 2) Q (2, 3) Q (3, 2) Q (2, 1) Q (1, 1) λ λ λ λ λ = λ. Qλ (3, 2) Qλ (2, 1) Qλ (1, 3) Qλ (3, 2) Qλ (2, 1)

This equality has a more compact interpretation on (time homogeneous) Markovian measure Pδ3,Qλ on {1, 2, 3}N where δ3 = (0, 0, 1) as 6
6
(3.1) Pδ3,Qλ [323211]1 = λPδ3,Qλ [321321]1 . Write a = 323211 ∈ {1, 2, 3}6 and b = 321321 ∈ {1, 2, 3}6 (words of length 6). The construction will k+6 rely on using elements in the groupoid of the tail relation which changes between the sequences [a]k k+6 and [b]k for some of the k’s according to the cylinder set. The following Lemma will play a role in ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 9 the inductive construction, to shorten notation write for z ∈ {a, b},

bN/6c X 6k Lz,N (x) = 1 6 ◦ T (x) = # {0 ≤ k ≤ bN/6c : x6k+1x6k+2..x6k+5 = z} . [z]1 k=0

n N N o The function Lz,n(x) is locally constant on cylinders of the form [x]1 : x ∈ {1, 2, 3} and hence N will be regarded as Lz,n : {1, 2, 3} → N.

Lemma 11. Let  > 0 and λ > 1. For any initial distribution π on {1, 2, 3} and p ∈ N there exists N ∈ N such that for all n > N,

Pπ,Qλ (x ∈ Σ: La,n(x) − Lb,n(x) > p) > 1 − .

 6N Proof. The measure Pπ,Qλ defines a measure on {1, 2, 3} which is a distribution of an aperiodic and irreducible Markov chain with state space S ⊂ {1, 2, 3}6. The result follows from the Mean ergodic 6
theorem for Markov chains, see for example [LPW, Th. 4.16], and the fact that, π ,Q [a] = P Qλ λ 1 6
6
λ π ,Q [b] > π ,Q [b] . P Qλ λ 1 P Qλ λ 1 

3.1. The inductive construction.

3.1.1. A short explanation of the idea behind the construction. The inductive construction inputs 3 ∞ sequences {λj,Mj,Nj}j=1 with M0 = 2 < N1, λk ↓ 1 and Mk  Nk+1  Mk+1 where an  bn K means an < bn for all n and an/bn → 0. Given {λj,Mj,Nj}j=1 we first choose λK+1 then NK+1 and then Mk+1. This choice of parameters defines a Markovian measure ν = M {πn,Pn : n ∈ Z} as follows: First for all n ≤ 1, P = Q = Q and π = π . The sequences {M ,N } gives rise to a partition n 1 n Q k k k∈N of N into two types of intervals Ik := [Mk−1,Nk) and Jk := [Nk,Mk). We set the transition matrices to be  Qλk , n ∈ Ik (3.2) Pn = Q, n ∈ (Z\N) ∪ (∪kJk) .

The measures πn are defined according to the consistency criteria (2.1). By the structure of the measure, the two sided shift is a K-automorphism with the one sided shift as its exact factor and the non singularity of the shift with respect to ν is only concerned with the Pn where Pn 6= Pn+1, that ∞ P∞ 2 is n ∈ ∪k=1 {Mk,Nk}. Non-singularity of the shift is thus equivalent to k=1 (λk − 1) < ∞ which follows from condition 1 on λk. Intuitively the first condition on λk enables one to approximate the Radon-Nikodym derivatives with respect to the shift by a finite product. The growth condition on P n 0 Mk − Nk then implies by the Hopf criteria that the shift is conservative since (T ) (x) = ∞ almost everywhere. The condition on Nk imply that ν is not equivalent to the stationary Markov measure M {πQ, Q}. This condition together with the second condition on λk enables us to show that the tail relation of the natural (noninvertible) factor of the Maharam extension is exact (by a reasoning which uses the EVC Lemma and the switching of [a]’s and [b]’s).

3.1.2. The construction. Induction base: Let λ1 be any number greater than 1, N1 be any natural number larger than M0 = 2 and M1 be any number greater than N1. ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 10

K k Induction step: Assume we are given {λk,Nk,Mk}k=1 ⊂ {(1, ∞) × N × N} . Notice that this defines πn and Pn for all n < MK by (3.2). Choose λK+1 > 1 so that (1) Finite approximation of the Radon-Nikodym derivatives condition:

1 2MK (3.3) (λK+1) < e 2K . (2) Lattice condition:

N (3.4) λK ∈ (λK+1) , where for x ∈ , xN = {xn : n ∈ }. Write ζ := log (λ ) ∈ . R N K+1 λK+1 1 N q Remark 12. By the Lattice condition for all k ≤ K + 1 there exists N 3 q ≤ ζK+1 so that λK+1 = λk.

Now that λK+1 is chosen, choose nK+1 large enough so that nK+1 > Mk and
(3.5) PπM ,Q x ∈ Σ: La,nK+1 (x) − Lb,nK+1 (x) > ζK+1 > 0.99. K λK+1

Notice that πMK is defined by PMK = QλK+1 and the consistency criteria 2.1 and that the existence of nK+1 follows from Lemma 11. Set NK+1 = MK + 6nK . Finally, choose MK+1 so that

−2NK+1 (3.6) (MK+1 − 2NK+1) λ1 ≥ 1. ∞ Theorem 13. Let {λk,Nk,Mk}k=1 be chosen according to the inductive construction and ν = M {πn,Pn : n ∈ Z} where πn and Pn are defined by (3.2). The shift (Σ, B, ν, T ) is a non-singular, conservative K- automorphism hence ergodic. Furthermore its Maharam extension is a K automorphism, hence it is stable type III1. The proof of Theorem 13 will be separated into two parts. We first show that the shift is a con- servative K-automorphism. Then in Subsubsection 3.1.3 we prove the K-property of the Maharam extension. Proof. [Non-Singularity,K property and conservativity]

Since ν ◦ T is the Markov measure generated by P˜j = Pj−1 and π˜j = πj−1, it follows from (2.2) and (3.2) that the shift is non-singular if and only if

∞ ( 2 X X q q  PNk (xNk , s) − PNk−1 (xNk , s) k=1 s∈S ) q q 2 + PMk (xMk , s) − PMk−1 (xMk , s) < ∞, µ ◦ T a.s. x.

G Since for all j ∈ Z, Pj (3, 2) ≡ 1,Pj (2, 1) = 1 − Pj (2, 3) ≡ 1+G , the sum is dominated by  2 ∞ (  2) ∞ s r ! X X q q X  λkG G 4 PNk (1, s) − PNk−1 (1, s) = 4 − 1 + λkG 1 + G k=1 s∈S k=1  !2 r 1 r 1  + − . 1 + λkG 1 + G  ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 11

∞ X 2 This sum converges or diverges together with |λk − 1| . As a consequence of condition (3.3) on k=1 {λj}, this sum is finite hence the shift is non-singular. Since Pj ≡ Q for all j ≤ 1, ∞ dµ ◦ T Y Pk−1 (xk, xk+1) T 0(x) = (x) = dµ Pk (xk, xk+1) k=1 ∞ is a BΣ+ measurable function. The sequence {Pj}j=1 satisfies (2.4) and hence the one sided shift 0 (Σ+, F, µ+, σ) is an exact factor such that T is F measurable. Here µ+ denotes the measure on the one sided shift space defined by {πj,Pj}j≥1. This shows that the two sided shift is a K automorphism. In order to show that the shift is conservative we first show that for all t ∈ N large enough and n ∈ [Nt,Mt − Nt), for almost every x ∈ Σ,

n 0 −2Nt (3.7) (T ) (x) ≥ λ1 /2, hence by (3.6), ∞ ∞ M −N X X Xt t (T n)0 (x) ≥ (T n)0 (x) = ∞.

n=1 t=1 n=Nt ±y By Hopf’s criteria, the shift is conservative. Fix t ∈ N and n ∈ [Nt,Mt − Nt). For x, y > 0, z = x denotes x−y ≤ z ≤ xy. By (2.3), n dµ ◦ T Y Pk−n (xk, xk+1) (x) = . dµ Pk (xk, xk+1) k∈N Pk−n6=Pk 0 First, we characterize the set of k s such that Pk−n 6= Pk. It follows from n ∈ [Nt,Mt − Nt) that t t ∪k=1 [Mk−1,Nk) − n ⊂ (−∞, 0] and ∪k=1 [Mk−1 + n, Nk + n) ⊂ [n, n + Nt) ⊂ [Nt,Mt). This means that for all u ≤ t, and k ∈ [Mu−1,Nu), Pk = Qλu and Pk−n = Q and for all k ∈ [Mu−1 + n, Nu + n),

Pk = Q and Pk−n = Qλu . One can check directly that for all other k ∈ [1,Mt), Pk = Pk−n = Q. ∞ Similarly for all k ≥ Mt, Pk−n = Pk, unless k ∈ ∪u=t+1 ([Nu,Nu + n) ∪ [Mu−1,Mu−1 + n)). For u > t, if k ∈ [Nu,Nu + n) then Pk = Q and Pk−n = Qλu and if k ∈ [Mu−1,Nu) then Pk = Q and

Pk−n = Qλu . Using this

n 0 (T ) (x) = It · I˜t where     t Nu Nu+n−1 Y Y Q (xk, xk+1) Y Qλu (xk, xk+1) It =   ·   Qλu (xk, xk+1) Q (xk, xk+1) u=1 k=Mu−1 k=Mu−1+n and M +n−1    ∞ u−1 Q (x , x ) Nu+n−1 Q (x , x ) ˜ Y Y k k+1 Y λu k k+1 It =   ·   . Qλu (xk, xk+1) Q (xk, xk+1) u=t+1 k=Mu−1 k=Nu For all u ∈ N and s, t ∈ {1, 2, 3} with Q(s, t) > 0, 1 Q (s, t) ≤ ≤ λu. λu Qλu (s, t) ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 12

Therefore     Mu−1+n−1 Nu+n−1 −2n Y Qλu (xk, xk+1) Y Q (xk, xk+1) 2n λu ≤   ·   ≤ λu , Q (xk, xk+1) Qλu (xk, xk+1) k=Mu−1 k=Nu

For all u > t, n < Mt ≤ Mu−1, thus ∞ ∞ Y Y (3.3) P∞ 1 ˜  ±2n  ±2Mu−1  ± n=t+1 n It = λu = λu = e 2 −−−→ 1. t→∞ u=t+1 u=t+1

Consequently there exists t0 ∈ N so that for all x ∈ ΣA, t > t0 and Nt ≤ n ≤ Mt − Nt, n 0 (T ) (x) ≥ It/2. Pt A similar analysis shows that (notice that λ1 = maxu∈N λu and u=1 (Nu − Mu−1) < Nt) t Y −2(Nu−Mu−1) −2Nt It ≥ λu ≥ λ1 , u=1 proving (3.7). 

3.1.3. Proof of the K-property of the Maharam extension. Denote by S :Σ+ → Σ+ the shift, π :Σ → + Σ+ the projection π(x) = (x1, x2, ··· ) := x+ and ν = ν|F where F = BΣ+ . The shift (Σ, B, ν, T ) is a 0 dν◦T K-automorphism and F is its exact factor. Since T (x) = dν (x) is F- measurable, the skew product 0 + −t

extension Slog T 0 (x+, y) = (Sx+, y + log T (x)) : Σ+ × R, F ⊗ BR, ν × e dt is well defined and the factor map π ×id(x, y) = (x+, y) is a factor map from the Maharam extension of T (denoted by T˜) ∞ ˜n to Slog T 0 . In addition F ⊗ BR is a minimal factor, that is ∨n=0T (F ⊗ BR) = BΣ ⊗ BR. It remains to show that Slog T 0 is an exact endomorphism. This is done by showing the ergodicity of the tail relation ϕˆ of Slog T 0 using its identification as T (S) where ϕˆ is the tail relation cocycle arising from the function ϕ = log T 0.3 Denote by −Mt−1 Ct := S {w ∈ Σ+ : La,nt (w) − Lb,nt (w) > ζt} .

(recall that nt = (Nt − Mt−1) /6). Notice that Ct is a finite union of cylinder sets of the form D = [w]Nt and that by (3.5) and the definition of ν (as an inhomogeneous Markov measure), Mt−1 ν+ (C ) = ({w ∈ Σ : L (w) − L (w) > ζ }) > 0.99. t PπMt−1 ,Qt + a,nt b,nt t Denote by C∗ the collection of all cylinder sets D = [d]Nt such that D ⊂ C . The following t Mt−1 t Mt−1−3 Fact\Lemma whose proof uses the specification of the SFT Σ+ is needed for gluing cylinders [c]1 ∗ with the cylinders in Ct .

n Lemma 14. For any C = [c]1 with n ≤ Mt−1 − 3 + −4 ν (C ∩ Ct) ≥ G (0.99)ν(C).

Proof. The claim uses the specification of Σ and goes by showing that for any cylinders of the form [c]Mt−1−3 and D = [d]Mt there exists a choice of a word w(c, d) = w w ∈ {1, 2, 3}2 such 1 Mt−1 Mt−1−2 Mt−1−1

3Note that ϕˆ is not necessarily the Radon- Nikodym cocycle of T (S). ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 13

cMt−1−3 w dMt−1 1 or 2 11 1 or 3 1 or 2 13 2 3 21 1 or 3 3 23 2 Table 1.

that


PMt−1−3 cMt−1−3, wMt−1−2 PMt−1−2 wMt−1−2, wMt−1−1 PMt−1−1 wMt−1−1, dMt−1 =


−4 Q cMt−1−3, wMt−1−2 Q wMt−1−2, wMt−1−1 Q wMt−1−1, dMt−1 ≥ G (see Table 1 for our choice of w). Writing wˆ(c, d) := cw(c, d)d for the concatenated word,

+  Nt  −4 +  Nt−Mt−1  −4 +  Nt−Mt−1  ν [w ˆ(c, d)]1 ≥ G ν (C)Pδd ,Qλ [d]1 ≥ G ν (C)PπM ,Qλ [d]1 . Mt−1 t t−1 t Summing over all the cylinders D = [d]Nt ∈ C∗, Mt−1 t X   ν+ (C ∩ C ) ≥ ν+ [w ˆ(c, d)]Nt ≥ G−4ν+ (C) (C ) ≥ (0.99)G−4ν+(C). t 1 PπMt−1 ,Qλt t ∗ D∈Ct n In the last inequality we used (3.5). If C = [c]1 with n < Mt−1 − 3 the result follows by writing Mt−1−3 {Mt−1−3}−n C = ]w[cw]1 where w ranges over all elements in {1, 2, 3} . 

n Lemma 15. Let t0 ∈ N. For every t0 ≤ t ∈ N and a cylinder C = [c]1 with n ≤ Mt−1 − 3 there exists V ∈ [[T (S)]] with Dom(V ), Ran(V ) ⊂ C, ν+ (Dom(V )) > (0.99) G−4ν(B) and for all w ∈ Dom(V ),

ϕˆ(w, V w) = − log λt0 .

ϕˆ Corollary 16. The tail relation T Slog T 0 = T (S) is ergodic.

n ∗ Proof. As before, let αn be the collection of n cylinders of the form [d]1 ∩ Σ+ and α = ∪nαn. The sequence αn is an increasing sequence of partitions of Σ+ which generates F. We first claim that ∞ {− log λt}t=1 ⊂ e (T (S), ϕˆ). Fix t0 ∈ N. By Lemma 15, for every t0, t ∈ N, t ≥ t0 and C ∈ αn + with n ≤ Mt−1 − 3, there exists V ∈ [[T (S)]] with (a) Dom(V ), Ran(V ) ⊂ C. (b) ν (Dom(V )) > −4 (0.99) G ν(B) and (c) for all w ∈ Dom(V ), ϕˆ(w, V w) = − log λt0 . By an application of the criteria from Lemma 7 with δ = (0.99) G−4 and C the algebra generated ∗ by α , − log λt0 ∈ e (T (S), ϕˆ). Since e (T (S), ϕˆ) is an additive subgroup of R, − log λt ↑ 0 and {− log λ } ⊂ e (T (S), ϕˆ), e (T (S), ϕˆ) = and thus T (S)ϕˆ is ergodic. t t∈N R 

n Proof of Lemma 15. Let t0, t ∈ N, t ≥ t0 and C = [c]1 with n ≤ Mt−1 − 3. By Remark 12 there exists 3 q = q((t, t ) ≤ ζ such that λq = λ . For D = [d]Mt ∈ C∗, there exists a minimal l = l(D) ≤ n N 0 t t t0 Nt t t such that for all w ∈ D

Mt−1
Mt−1
Mt−1
Mt−1
La,l S w − Lb,l S w = La,l S d − Lb,l S d = q. ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 14

Write D(i) = [d]Mt−1+6(i+1), 0 ≤ i ≤ l(D) and for convenience write D = D(0)D(1) ··· D(l(D))[d]Nt . Mt−1+6i Mt−1+6l(D) h iNt h iNt Let Dˆ = dˆ be defined by the following rule. Firstly dˆ = [d]Nt . For Mt−1+6l(D) Mt−1 Mt−1+6l(D) 1 ≤ i ≤ l(D),  D(i),D(i) ∈/ {a, b}  Dˆ(i) = b D(i) = a  a D(i) = b. ˆ ˆ The cylinder D satisfies dMt−1 = dMt−1 and

Mt−1 Mt−1 La,l(S w) Lb,l(S w) + (a) ! (b) ! ν (D) Pδ3,Qλt Pδ3,Qλt (3.1) q −1 (3.8) = = λt = λt . +   (b) (a) 0 ν Dˆ Pδ3,Qλt Pδ3,Qλt

Similarly, since Q = Q1,

Nt−1 Pδd ,Q (D) Mt−1 Y Q (dk, dk+1) (3.9)   =   = 1. Dˆ Q dˆ , dˆ Pδdˆ ,Q k=Mt−1 k k+1 Mt−1

This defines an element map VD : D → Dˆ in [[T (S)]] by taking a point w = wdw˜ with w ∈ Mt−1−1 ˆ {1, 2, 3} and w˜ ∈ Σ+ to VD(w) = wdw˜ (that is changing only the values of w in the coor- dinates of the cylinder set D). First we claim that for all w ∈ D,

ϕˆ (w, VD(w)) = − log λt0 .

Nt Nt To see this notice first that S w = S VD(w) for all w ∈ D, hence by (2.3) and (2.5), dν ◦ T Nt dν ◦ T Nt ϕˆ (w, V (w)) = log (w) − log (V (w)) D dν dν D   Dˆ + Pδdˆ ,Q ν (D) Mt−1 = log + log = − log λt . + ˆ 0 ν (D) Pδd ,Q (D) Mt−1 ∗ Set V : Ct → Σ+ by for D ∈ Ct , V |D = VD. By the previous argument for all w ∈ Ct, ϕˆ(w, V w) = ∗ − log λt0 . We claim that V ∈ [[T (S)]] with Dom(V ) = Ct. Since for all w ∈ D ∈ Ct , (w, V (w)) = (w, VD(w)) ∈ T (S) it remains to show that V is one to one which follows from the injectivity of the ˆ ∗ map D 7→ D : Ct → F. The injectivity of the latter is established by looking at the position of the a’s ˆ ˆ h iMt−1+6l(D) and b’s in dˆ where Mt−1   n     o ˆ Mt−1 ˆ Mt−1 ˆ l Dˆ = min k ≤ nt : Lb,k S d − La,k S d = q

and noticing that the similar operation of the definition of dˆ from D with l(D) replaced by ˆl(D) + −4 + recovers D. By Lemma (14), ν (C ∩ Ct) > (0.99) G ν (C) and the element V |C∩Ct ∈ [[T (S)]] satisfies the conclusion of the Lemma. 

N Remark 17. The countable group S∞ of all finite permutations of N acts on {1, 2, 3} via π(w)n = wπn for π ∈ S∞ and w ∈ {1, 2, 3}N. This action generates the exchangeability relation. In the proof of ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 15

∗ Lemma 15, for all D ∈ Ct , we specified π ∈ S∞ such that VD = π|D and thus V is locally constant and + chosen from the orbit of S∞ on Σ+. Since µ = M {πQ, Q} is a Gibbs measure, in fact the measure of maximal entropy on Σ+, it is an invariant measure for the action of S∞, see [ANS]. By this, dν+ ◦ π| ϕˆ(w, V w) = − log D (w). D dν+

By a more involved argument using these permutations, one can show that if Pn = Qλn with λn = 1 for every n ≤ 1 and ν = M {πn,Pn}, if ν is singular with respect to M {πQ1 , Q1} and the shift is ν-nonsingular then the shift (Σ, ν, T ) is stable type III1. See [DaL, Kos2] for some recent results of this type in the case of 2 state Markov Chains and half-stationary Bernoulli shifts.

1 d 4. Stable type III1 Anosov C Diffeomorphims of T , d ≥ 3. Let (M, d) be a Riemannian manifold. A diffeomorphism f : M → M is an Anosov diffeomorphism x x if there exists λ > 1, C > 0 and a decomposition TxM = Es ⊕ Eu, x ∈ M, such that: s s u • the decomposition is Df-equivariant. That is for all x ∈ M, Df (x)Ex = Ef(x) and Df (x)Ex = u Ef(x). s n • for all x ∈ M, v ∈ Ex and n ∈ N, |Df n (x)v| ≤ Cλ |v| (contraction on the stable manifolds). u n • for all x ∈ M, v ∈ Ex and n ∈ N, Df −n (x)v ≤ Cλ |v| (expansion on the unstable manifolds). 2 2 Example. f : T → T the toral automorphism defined by ! 1 1 x f(x, y) = ({x + y}, x) = mod 1, 1 0 y ! 1 1 where {t} is the fractional part of t. Since det = 1, f preserves the Lebesgue measure on 1 0 2 2 T . In addition for every z ∈ T , the tangent space can be decomposed as span {vs} ⊕ span {vu} where vu = (1, 1/G) and vs = (1, −G) .

2 For every w ∈ Vu := span {vu} and x ∈ T ! 1 1 Df (x)w = w = Gw, 1 0

2 1
2 s s For every u ∈ Vs := span {vs} and x ∈ T ,Df (x)u = − G u. In this example, for all x ∈ T , Ex = V u u and Ex = V . These facts can be used [Adl, AW] to construct the Markov partition for f with three elements {R1,R2,R3}, see figure 4.1. The adjacency Matrix of the Markov partition is then defined by Ai,j = 1 if and only if Ri ∩ −1 f (Rj) 6= ∅. Here the adjacency Matrix is   1 0 1   A =  1 0 1  . 0 1 0

2 Let Φ : Σ = ΣA → T be the map defined by ∞ \ −n Φ(x) := f Rxn , n=−∞ ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 16

vs R3

vu

R1

R2

Figure 4.1. The construction of the Markov partition

∞ n N −n o note that ∩n=−N f Rxn is a decreasing sequence of compact sets hence by the Baire Category N=1 2 Theorem Φ(x) is well defined. The map Φ:Σ → T is continuous, finite to one, and for every x ∈ ΣA, Φ ◦ T (x) = f ◦ Φ(x).

2 3 −n −1 In addition, for every x ∈ T \ ∪n∈Z ∪i=1f (∂Ri) there exists a unique w ∈ Σ so that w = Φ (x). 2
Thus Φ is a semi-conjugacy (topological factor map) between (ΣA,T ) to T , f . The Lebesgue 2 3 −n measure λ on T is conservative and invariant under f and the set ∪n∈Z ∪i=1 f (∂Ri) (points with non-unique expansion in Σ) is a Lebesgue-null set. As a consequence Φ is an isomorphism between
2
ΣA, µπQ,Q,T and T , λ, f where µπQ,Q is the stationary Markov measure with  G 1  1+G 0 1+G (4.1) P ≡ Q :=  G 0 1  j  1+G 1+G  0 1 0 and  √    1/ 5 λ (R ) √ 1 (4.2) π = π :=  1/G 5  =  λ (R )  . j Q  √   2  1/G 5 λ (R3)

Similarly, letting ν = M {πn,Pn : n ∈ Z} be a measure arising from the inductive construction, ν is nonatomic, the shift (Σ, ν, T ) is conservative and Φ is one to one on the support of µ. Thus 2
2
Φ : (Σ, BΣ, ν, T ) → T , BT2 , µ = Φ∗ν, f is an isomorphism. Therefore T , BT2 , µ = Φ∗ν, f is stable type III1 .The modified inductive construction of [Kos1] inserts another sequence k ↓ 0 in the inductive construction via the following order (In Subsection (4.1) we specify the modified induction process),

K {λj,Nj, , j,Mj}j=1 → λK+1 → NK+1 → K+1 → MK+1. 2 2 The construction of [Kos1] then defines an homeomorphism h : T → T such that

• (h)∗ λ ∼ Φ∗ν where ν = M {πn,Pn : n ∈ Z} is the measure coming from the inductive con- 2
struction. By Fact 10, T , BT2 , (h)∗ λ, f is stable type III1. ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 17

−1 2 2 1 • g =h ◦ f ◦ h : T → T is a C Anosov diffeomorphism.

2
1 Theorem 18. (i) T , BT2 , λ, g is a C Anosov diffeomorphism which is stable type III1. d d d+2
(ii) For any hyperbolic linear toral automorphism Q : T → T the transformation g×Q : T , BTd+2 , λd+2 d+2 2 d is stable type III1 where λd+2 is the Lebesgue measure on T and for x ∈ T , y ∈ T , g × Q (x, y) = (g(x),Q(y)).

2
2
Proof. T , BT2 , λ, g is stable type III1 since it is isomorphic to T ,, BT2 , (h)∗ λ, f (via the homeo- morphism h). Every hyperbolic toral automorphism is mixing with respect to the Lebesgue measure d on T hence weak mixing. Part (ii) follows from the proof of the second part of Corollary 2. 

Remark 19. The class of Riemmanian manifolds M for which there exists an Anosov diffeomorphism Q : M → M are called Anosov Manifolds. It is a famous open question whether every Anosov Manifold is an infranil manifold, see [Gor] and the references therein. If M is an Anosov manifold such that 4 there exists an Anosov diffeomorphism Q : M → M which preserves a measure µ  volM , then g×Q 2 2 is a stable type III1 Anosov diffeomorphism of T × M with respect to the volume measure on T × M. Since there are no Anosov diffeomorphims of T, the question whether there are type III1 Anosov 3 diffeormorphisms of T is still open. The main challenge in this case is that the Markov partitions of 3 hyperbolic toral automorphism of T have fractal boundaries. This presents further difficulties in the smooth realization process (construction of h).

4.1. The modified induction process (inserting { }∞ ): We first begin by warning the reader √ t t=1 1+ 5 that in [Kos1], ϕ = G = 2 , hence in all statements there which involve ϕ one should swap ϕ to G. 2−N1 The modified induction process is as follows. First we demand that N1 > 20 and that λ1 < e . As this involves only a change in the basis of the inductive construction, such choices are possible.

Choose 1 small enough in order to satisfy the conditions of [Kos1, Proposition 8.2] (with k = 1). Finally let M1 be large enough to satisfy the conditions of [Kos1, Lemma 9.1]. K Given {λj,Nj, j.Mj}j=1 we first choose λK+1 to satisfy

−N 2MK 2 K λK+1 ≤ e , and the lattice condition (3.4). This condition is compatible with the conditions on λK+1 in Section 3. Secondly, choose NK+1 to satisfy (3.5). That is the conditions on λK+1 and NK+1 do not depend on ’s.

Now choose K+1 to be small enough so that the conclusions of [Kos1, Prop. 8.2, Lemmas 9,10,11] hold for k = K + 1.

Finally choose MK+1 large enough with respect to NK+1 to satisfy (3.6) and the conclusion of [Kos1, Lemma 9.1]. This finishes the inductive step.

Acknowledgement. I would like to thank the referee for the remarks and corrections which have significantly improved the paper. This research was supported in part by a Eurpoean Advanced Grant StochExtHom (ERC AdG 320977).

4 volM is the volume measure on M ON MANIFOLDS ADMITTING STABLE TYPE III1 ANOSOV DIFFEOMORPHISMS 18

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